By varying \(\varepsilon \) , we can study how the shape of our point-cloud data evolves across scales. Specifically, we are interested in the presence of \(k\) -dimensional holes, meaning empty regions bound by \(k\) -simplices [1]}, [2]}. Connected components are 0-dimensional holes, and 2D trapped areas are 1-dimensional holes (i.e., loops) bound by 1-simplices. As we allow \(\varepsilon \)  to grow, the shape of our data in dimension 0 evolves from \(N\)  isolated points when \(\varepsilon = 0\)  to one large connected component once \(\varepsilon \)  is large enough. McGuirl et al. [3]} interpreted dimension-1 topological features, in particular, as holding information about stripe patterns generated by the ABM [4]}. Because the ABM [4]} has periodic boundary conditions in \(x\) , pattern formation is occurring on a cylinder, as we show in Figure REF (b). If we apply persistent homology to the positions of \(X^\text{d}\)  cells on this cylindrical domain, interstripes are visible as loops in our data for a range of \(\varepsilon \)  values. On the other hand, applying persistent homology to the positions of \(M\)  or \(X^\ell \)  cells translates into information about stripes [3]}.
